Chapter Four

1
Chapter Four

• Utility

2
Utility Functions

• Want to come up with a compact way of
  representing preferences.
• A Utility function is a way of assigning a number
  to every possible consumption bundle such that
  more preferred bundles get assigned larger
  numbers.

3
Utility Functions

• A utility function U(x) represents a preference
  relation if and only if for any two bundles x
  and x x x U(x)
  gt U(x) x x U(x) lt
  U(x) x x U(x)
  U(x).

p
p
4
Utility Functions

• Example Suppose there are two goods x1 and x2.
  The consumer is indifferent between the bundle
  (4, 1) and (2,2) but the bundle (2,3) is strictly
  preferred to both of these.
• The following utility function will represent
  this preference ordering.
• U(x1, x2) x1 x2
• Then U(4,1)4 and U(2,2)4 and U(2,3)6.
• These numbers are called Utility Levels.

5
Utility Functions Indiff. Curves

• An indifference curve contains equally preferred
  bundles.
• Equal preference bundles must have the same
  utility level.
• Therefore, all bundles on an indifference curve
  have the same utility level.

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Utility Functions Indiff. Curves

• So the bundles (4,1) and (2,2) are on the indiff.
  curve with utility level U º 4
• But the bundle (2,3) is in the indiff. curve with
  utility level U º 6.
• On an indifference curve diagram, this preference
  information looks as follows

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Utility Functions Indiff. Curves
x2
(2,3) (2,2) (4,1)
p
U º 6
U º 4
x1
8
Utility Functions Indiff. Curves

• Another way to visualize this same information is
  to plot the utility level on a vertical axis.
• U(x1,x2) is a three dimensional surface.

9
Utility Functions Indiff. Curves
3D plot of consumption utility levels for 3
bundles
U(2,3) 6
Utility
U(2,2) 4 U(4,1) 4
x2
x1
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Utility Functions Indiff. Curves

• This 3D visualization of preferences can be made
  more informative by adding more possible
  consumption bundles.
• We can also draw indifference curves by
  connecting all possible bundles that yield the
  same level of utility.

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Utility Functions Indiff. Curves
Utility
U º 6
U º 5
U º 4
U º 3
x2
U º 2
U º 1
x1
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Utility Functions Indiff. Curves

• Comparing all possible consumption bundles gives
  the complete collection of the consumers
  indifference curves, each with its assigned
  utility level.
• This complete collection of indifference curves
  completely represents the consumers preferences.

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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
U
x2
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map

• When we typically draw indiffernce curves, the
  drawing is in 2 dimensions.
• Really what we are doing is projecting the
  indifference curves on the x1/x2 plane.
• One can think of this drawing as being like a
  topographical map of utility.

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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
20
Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
21
Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x1
22
Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
25
Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
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Utility Functions Indiff. CurvesIndifference
Curves are like a topographic map
x2
x1
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Utility Functions Indiff. Curves

• The collection of all indifference curves for a
  given preference relation is an indifference map.

29
Utility Functions Indiff. Curves
x2
U º 6
U º 4
U º 2
x1
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Utility Functions and Their Indifference Curves

• How do we get indifference curves from utility
  functions?
• We can always figure out what the indifference
  curves look like by finding all the points that
  yield the same level of utility.

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Example 4.1 Utility Functions and Their
Indifference Curves

• Example 4.1 Consider the utility function
  U(x1,x2) x1x2 .
• Draw the indifference curve associated with a
  utility level of 100.
• Find all points such that x1x2100 or all points
  where x2100/x1.
• From here we can just plot some points for
  example when x1 1, x1 10, x1 100, etc.

32
Example 4.1 Utility Functions and Their
Indifference Curves
x2
100

10
U100
1
x1
1
10
100
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Utility Functions and Their Indifference Curves

• We now look at several types of utility functions
  and derive how their indifference curves look.
• Perfect Substitutes
• Perfect Complements
• Quasi-Linear
• Cobb-Douglas

34
Example 4.2 Utility Functions and Their
Indifference Curves Perfect Substitutes

• Consider the Utility Function
  V(x1,x2) x1 x2.
• What do the indifference curves for this perfect
  substitution utility function look like?
• Draw the indifference curves associated with
  utility levels of 5, 9, and 13.

35
Example 4.2 Utility Functions and Their
Indifference Curves Perfect Substitutes
x2
x1 x2 5
13
x1 x2 9
9
x1 x2 13
5
V(x1,x2) x1 x2.
5
9
13
x1
Perfect Substitute Indifference Curves are
linear and parallel.
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Utility Functions and Their Indifference Curves
Perfect Substitutes

• Any utility function of the form V(x1,x2)
  Ax1 Bx2 where A and B are positive constants
  has indifference curves that are
• Parallel and Linear

37
Example 4.3 Utility Functions and Their
Indifference Curves Perfect Complements

• Now consider the utility function W
• W(x1,x2) minx1,x2.
• What do the indifference curves for this perfect
  complementarity utility function look like?
• Draw the indifference curves associated with the
  utility levels of 3, 5, and 8.

38
Example 4.3 Utility Functions and Their
Indifference Curves Perfect Complements
x2
45o
W(x1,x2) minx1,x2
minx1,x2 8
8
minx1,x2 5
5
3
minx1,x2 3
3
5
8
x1
All indifference curves are right-angled
with vertices on a ray from the origin.
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Utility Functions and Their Indifference Curves
Perfect Complements

• Any utility function of the form
  W(x1,x2) minAx1,Bx2 where A and B are some
  positive constants will have indifference curves
  that are
• right-angled
• with vertices on a ray from the origin.

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Utility Functions and Their Indifference Curves
Quasi-linear

• A utility function of the form
  U(x1,x2) f(x1) x2(where f(x1) is some
  strictly concave function) is linear in just x2
  and is called quasi-linear.
• E.g. U(x1,x2) 2x11/2 x2.

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Utility Functions and Their Indifference Curves
Quasi-linear

• The general form for the indifference curve with
  utility level k for the utility function of the
  form U(x1,x2) f(x1) x2 is

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Utility Functions and Their Indifference Curves
Quasi-linear
x2
Each curve is a vertically shifted copy of the
others. These indifference curves may
touch both axes.
x1
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Utility Functions and Their Indifference Curves
Cobb-Douglas

• Any utility function of the form
  U(x1,x2) x1a x2bwith a gt 0 and b gt 0 is
  called a Cobb-Douglas utility function.
• E.g. U(x1,x2) x11/2 x21/2 (a b 1/2)
  V(x1,x2) x1 x23 (a 1, b 3)
• W(x1,x2) x1 x2 (a 1,
  b 1)

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Utility Functions and Their Indifference Curves
Cobb-Douglas

• The general form for the indifference curve with
  utility level k for the utility function of the
  form U(x1,x2) x1a x2b is

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Utility Functions and Their Indifference Curves
Cobb-Douglas
x2
All Cobb-Douglas indifference curves are
strictly convex, and asymptoting to but
nevertouching any axis.
x1
46
Marginal Utilities

• We now look at what is meant by marginal utility
  and how to use this concept to derive an equation
  for the slope of indifference curves.

47
Marginal Utilities

• Marginal means additional or incremental
  change.
• The marginal utility of commodity i is the
  rate-of-change of total utility as the quantity
  of commodity i consumed changes (ceteris
  paribus) i.e.

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Example 4.4 Marginal Utilities

• E.g. if U(x1,x2) x11/2 x22 then

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Example 4.4 Marginal Utilities

• E.g. if U(x1,x2) x11/2 x22 then

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Example 4.4 Marginal Utilities

• E.g. if U(x1,x2) x11/2 x22 then

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Example 4.4 Marginal Utilities

• E.g. if U(x1,x2) x11/2 x22 then

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Example 4.4 Marginal Utilities

• So, if U(x1,x2) x11/2 x22 then

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Marginal Utilities Goods, Bads and Neutrals

• A good is a commodity which increases utility
  (gives a more preferred bundle when increased).
• A bad is a commodity which decreases utility
  (gives a less preferred bundle when increased).
• A neutral is a commodity which does not change
  utility (gives an equally preferred bundle when
  changed).

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Goods, Bads and Neutrals

• For a good, Marginal Utility (MU)gt0
• For a bad, MUlt0
• For a neutral, MU0

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Marginal Utilities and Marginal
Rates-of-Substitution

• We can use the Utility function and marginal
  utilities to get a formulation for the Marginal
  Rate of Substitution (MRS)
• Recall, the MRS slope of the indifference
  curve.
• In the two good case, MRSdx2/dx1
• How do we get this from the utility function?

56
Marginal Utilities and Marginal
Rates-of-Substitution

• The general equation for an indifference curve
  is U(x1,x2) º k, for k a constant.
• (i.e. All points that yield the exact same level
  of utility of k units).

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Marginal Utilities and Marginal
Rates-of-Substitution

• Along an indifference curve U(x1,x2) º k
• Totally differentiating this identity gives

58
Marginal Utilities and Marginal
Rates-of-Substitution
rearranged is
59
Marginal Utilities and Marginal
Rates-of-Substitution
Or solving for dx2/dx1
This is the MRS.
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Marginal Utilities and Marginal
Rates-of-Substitution

• In other words
• MRS - Marginal utility of good 1 divided by the
  Marginal utility of good 2

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Example 4.5 Marg. Utilities Marg.
Rates-of-Substitution

• Suppose U(x1,x2) x1x2.
• Calculate the MRS at the points (1,8) and (6,6).
• Draw the associated indifference curves.

62
Example 4.5 Marg. Utilities Marg.
Rates-of-Substitution

• Suppose U(x1,x2) x1x2. Then

so
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Example 4.5 Marg. Utilities Marg.
Rates-of-Substitution
MRS(1,8) - 8/1 -8MRS(6,6) - 6/6 -1. Are
these points on the same indifference Curve?
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Example 4.5 Marg. Utilities Marg.
Rates-of-Substitution
U(x1,x2) x1x2
x2
8
MRS(1,8) - 8/1 -8 MRS(6,6) - 6/6
-1.
6
U 36
U 8
x1
1
6
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Marg. Rates-of-Substitution for Quasi-linear
Utility Functions

• A quasi-linear utility function is of the form
  U(x1,x2) f(x1) x2.

so
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Marg. Rates-of-Substitution for Quasi-linear
Utility Functions

• MRS - f (x1) does not depend upon x2 (the
  linear variable) so the slope of indifference
  curves for a quasi-linear utility function is
  constant along any line for which x1 is constant.
  
• What does that make the indifference map for a
  quasi-linear utility function look like?

67
Marg. Rates-of-Substitution for Quasi-linear
Utility Functions
x2
Each curve is a vertically shifted copy of the
others.
MRS -f(x1)
MRS is a constantalong any line for which x1
isconstant.
x1
x1
x1
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Utility Functions and Monotonic Transformations

• Utility is an ordinal (i.e. ordering) concept.
• For example, consider two bundles
• A (a1, a2) and B(b1, b2).
• If U(a1, a2) 6 and U(b1, b2) 2 then bundle A is
  strictly preferred to bundle B. But it does not
  imply that A is preferred three times as much as
  is B.
• This implies that there are many possible utility
  functions for a given set of preferences.

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Utility Functions Monotonic Transformations

• There is no unique utility function representing
  a preference relation.
• For example, the following utility functions
  represent the same preferences
• U(x1,x2) x1x2
• VU2 x12x22

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Utility Functions Monotonic Transformations

• U(x1,x2) x1x2, soU(2,3) 6 gt U(4,1) U(2,2)
  4that is, (2,3) (4,1) (2,2).
• V(x1,x2 ) x12x22
• V(2,3) 36gt U(4,1) U(2,2) 16that is,
  (2,3) (4,1) (2,2).
• V preserves the same order as U and so represents
  the same preferences.

p
p
71
Utility Functions Monotonic Transformations
p

• U(x1,x2) x1x2 (2,3) (4,1)
  (2,2).
• We could also define another utility function W
  where
• W 2U 10 2(x1x2) 10

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Utility Functions Monotonic Transformations
p

• U(x1,x2) x1x2 (2,3) (4,1)
  (2,2).
• Define W 2U 10.
• Then W(x1,x2) 2x1x210 so W(2,3) 22 gt
  W(4,1) W(2,2) 18. Again,(2,3) (4,1)
  (2,2).
• W preserves the same order as U and V and so
  represents the same preferences.

p
73
Utility Functions Monotonic Transformations

• In the previous examples, V and W are what are
  called (positive) monotonic transformations of U.
• A positive monotonic transformation f(U)
  transforms each number U in a way that preserves
  the order of the numbers such that if U1gtU2, then
  f(U1)gtf(U2)

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Utility Functions Monotonic Transformations

• If
• U is a utility function that represents a
  preference relation and
• f is a strictly increasing function,
• then V f(U) is also a utility
  functionrepresenting .

75
Monotonic Transformations Marginal
Rates-of-Substitution

• Applying a monotonic transformation to a utility
  function representing a preference relation
  simply creates another utility function
  representing the same preference relation.
• What happens to marginal rates-of-substitution
  when a monotonic transformation is applied?

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Monotonic Transformations Marginal
Rates-of-Substitution

• For U(x1,x2) x1x2 the MRS - x2/x1.
• Create V U2 i.e. V(x1,x2) x12x22. What is
  the MRS for V?which is the same as the MRS
  for U.

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Monotonic Transformations Marginal
Rates-of-Substitution

• More generally, if V f(U) where f is a strictly
  increasing function, then

So MRS is unchanged by a positivemonotonic
transformation.
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Utility Functions Monotonic Transformations

• How do you tell if a function is a monotonic
  transformation of another?
• One way is to check to see if the slopes of the
  indifference curves, (the MRSs) are the same.

79
Example 4.6 Utility Functions/ Monotonic
Transformations

• Example Are the following utility functions V
  and W (positive) monotonic transformations of one
  another?
• V(x1, x2)x1x2
• W(x1, x2) ln(x1) ln(x2)
• Show by calculating the MRS for each.

80
Example 4.6 Utility Functions/ Monotonic
Transformations

• MRS of utility function V is given by
• MRS of utility function W is given by
• Since the MRSs are the same, the utility
  functions represent the same preferences.